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受欢迎的微积分 >

integral of (sin(x/2))/(cos^2(x/2))

解答

\int \frac{sin ( \frac{x}{2} )}{cos ^{2} ( \frac{x}{2} )} d x

解答

2 sec (\frac{x}{2}) + C

求解步骤

\int \frac{sin ( \frac{x}{2} )}{cos ^{2} ( \frac{x}{2} )} d x

使用三角恒等式改写

= \int sec (\frac{x}{2}) tan (\frac{x}{2}) d x

使用换元积分法

= \int sec (u) tan (u) \cdot 2 d u

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int sec (u) tan (u) d u

使用换元积分法

= 2 \cdot \int 1 d v

常数积分:

\int a d x = a x

= 2 \cdot 1 \cdot v

代回

= 2 \cdot 1 \cdot sec (\frac{x}{2})

化简

= 2 sec (\frac{x}{2})

解答补常数

= 2 sec (\frac{x}{2}) + C

作图

流行的例子

derivative f(x)=2x-6

d er i v a t i v e f (x) = 2 x - 6

d/(dt)(e^t+e^{-t})

\frac{d}{d t} (e^{t} + e^{- t})

limit as x approaches 3 of (x+4)/(x-2)

x \to 3 lim (\frac{x + 4}{x - 2})

integral of (2cos(x)-3sec(x)tan(x))

\int (2 cos (x) - 3 sec (x) tan (x)) d x

derivative of ((sin(x)/(1+cos(x)))^2)

\frac{d}{d x} ((\frac{sin ( x )}{1 + cos ( x )})^{2})